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Mirrors > Home > MPE Home > Th. List > sbccsb2 | Structured version Visualization version GIF version |
Description: Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.) (Revised by NM, 18-Aug-2018.) |
Ref | Expression |
---|---|
sbccsb2 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3586 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
2 | elex 3352 | . 2 ⊢ (𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑} → 𝐴 ∈ V) | |
3 | abid 2748 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
4 | 3 | sbcbii 3632 | . . 3 ⊢ ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑) |
5 | sbcel12 4126 | . . . 4 ⊢ ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ⦋𝐴 / 𝑥⦌𝑥 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}) | |
6 | csbvarg 4146 | . . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) | |
7 | 6 | eleq1d 2824 | . . . 4 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝑥 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑} ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})) |
8 | 5, 7 | syl5bb 272 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})) |
9 | 4, 8 | syl5bbr 274 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})) |
10 | 1, 2, 9 | pm5.21nii 367 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∈ wcel 2139 {cab 2746 Vcvv 3340 [wsbc 3576 ⦋csb 3674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-nul 4059 |
This theorem is referenced by: (None) |
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